Concept
Catalan number
Related concepts (16)
Enumerative combinatorics
Enumerative combinatorics is an area of combinatorics that deals with the number of ways that certain patterns can be formed. Two examples of this type of problem are counting combinations and counting permutations. More generally, given an infinite collection of finite sets Si indexed by the natural numbers, enumerative combinatorics seeks to describe a counting function which counts the number of objects in Sn for each n.
Integer sequence
In mathematics, an integer sequence is a sequence (i.e., an ordered list) of integers. An integer sequence may be specified explicitly by giving a formula for its nth term, or implicitly by giving a relationship between its terms. For example, the sequence 0, 1, 1, 2, 3, 5, 8, 13, ... (the Fibonacci sequence) is formed by starting with 0 and 1 and then adding any two consecutive terms to obtain the next one: an implicit description. The sequence 0, 3, 8, 15, ... is formed according to the formula n2 − 1 for the nth term: an explicit definition.
Narayana number
In combinatorics, the Narayana numbers form a triangular array of natural numbers, called the Narayana triangle, that occur in various counting problems. They are named after Canadian mathematician T. V. Narayana (1930–1987). The Narayana numbers can be expressed in terms of binomial coefficients: The first eight rows of the Narayana triangle read: An example of a counting problem whose solution can be given in terms of the Narayana numbers , is the number of words containing n pairs of parentheses, which are correctly matched (known as Dyck words) and which contain k distinct nestings.
Combinatorial class
In mathematics, a combinatorial class is a countable set of mathematical objects, together with a size function mapping each object to a non-negative integer, such that there are finitely many objects of each size. The counting sequence of a combinatorial class is the sequence of the numbers of elements of size i for i = 0, 1, 2, ...; it may also be described as a generating function that has these numbers as its coefficients. The counting sequences of combinatorial classes are the main subject of study of enumerative combinatorics.
Dyck language
In the theory of formal languages of computer science, mathematics, and linguistics, a Dyck word is a balanced string of brackets. The set of Dyck words forms a Dyck language. The simplest, D1, use just two matching brackets, e.g. ( and ). Dyck words and language are named after the mathematician Walther von Dyck. They have applications in the parsing of expressions that must have a correctly nested sequence of brackets, such as arithmetic or algebraic expressions. Let be the alphabet consisting of the symbols [ and ].
Associahedron
In mathematics, an associahedron K_n is an (n – 2)-dimensional convex polytope in which each vertex corresponds to a way of correctly inserting opening and closing parentheses in a string of n letters, and the edges correspond to single application of the associativity rule. Equivalently, the vertices of an associahedron correspond to the triangulations of a regular polygon with n + 1 sides and the edges correspond to edge flips in which a single diagonal is removed from a triangulation and replaced by a different diagonal.
Permutation pattern
In combinatorial mathematics and theoretical computer science, a permutation pattern is a sub-permutation of a longer permutation. Any permutation may be written in one-line notation as a sequence of digits representing the result of applying the permutation to the digit sequence 123...; for instance the digit sequence 213 represents the permutation on three elements that swaps elements 1 and 2.
Partition of a set
In mathematics, a partition of a set is a grouping of its elements into non-empty subsets, in such a way that every element is included in exactly one subset. Every equivalence relation on a set defines a partition of this set, and every partition defines an equivalence relation. A set equipped with an equivalence relation or a partition is sometimes called a setoid, typically in type theory and proof theory. A partition of a set X is a set of non-empty subsets of X such that every element x in X is in exactly one of these subsets (i.
Binomial transform
In combinatorics, the binomial transform is a sequence transformation (i.e., a transform of a sequence) that computes its forward differences. It is closely related to the Euler transform, which is the result of applying the binomial transform to the sequence associated with its ordinary generating function. The binomial transform, T, of a sequence, {an}, is the sequence {sn} defined by Formally, one may write for the transformation, where T is an infinite-dimensional operator with matrix elements Tnk.
Noncrossing partition
In combinatorial mathematics, the topic of noncrossing partitions has assumed some importance because of (among other things) its application to the theory of free probability. The number of noncrossing partitions of a set of n elements is the nth Catalan number. The number of noncrossing partitions of an n-element set with k blocks is found in the Narayana number triangle. A partition of a set S is a set of non-empty, pairwise disjoint subsets of S, called "parts" or "blocks", whose union is all of S.